Bondora Credit Allocation Study

Progress Meeting 2

Group 7

Jheronimus Academy of Data Science

How are Different Kinds of Credit Distributed among Bondora Borrowers?

Background & Premise

  • Bondora P2P Lending Service
  • European dataset spanning between 2009-2021 with 179,000+ individual loans
  • Data on 112 attributes of users individually and the characteristics of their loans

Network Information

Type

\(\mathrm{B}_1\)

\(\mathrm{B}_2\)

Undirected Bipartite

Borrowers identified by UserName

Reported LoanUse
Edges

\(\mathrm{B}_1 \rightarrow \mathrm{B}_2:\) User takes out specific kind of loan

\(\mathrm{B}_2 \rightarrow \mathrm{B}_1:\) Specific loan type belongs to a User

\(\mathrm{B}_1 \nleftrightarrow \mathrm{B}_2:\) Forbidden edges

Research Questions & Hypotheses

  • How do loan characteristics affect the frequency credit allocation among borrowers?

  • Studied using QAP Linear Regressions since adjacency matrices are valued

Hypothesis Expected Result Specific Method
H1a : Users with the same loan use are likely to have the same credit rating \(\hat{\beta}_{1,1}>0\)

\(Y=\) Adjacency matrix of shared loan use

\(X_1=\) Adjacency matrix of shared credit rating

\(+\) \(X_{2,3,..,n}=\) Controls

  • How is the structure credit type allocation affected by borrower behaviour and its characteristics?

  • Studied using Bipartite ERGMs with binary edges

Hypothesis Expected Result ERGM Term
H2a : Pairs of borrowers share same loan type \(\hat{\theta}>0\)

Endogenous + Dyad-dependent

gwb1dsp()
H3a : Borrowers have a non-diverse loan use mix \(\hat{\theta}<0\)

Endogenous + Dyad-dependent

gwb1degree()
H4a : Younger borrowers have a different loan use mix \(\hat{\theta}\neq0\)

Exogenous + Dyad-independent

b1cov("b1_age")
H5a : Borrowers of differing genders have different loan use mixes \(\hat{\theta}<0\)

Heterophily + Exogenous + Dyad-independent

b1nodematch("b1_gender")

Note: measured indirectly through heterophily because nodemix() does not have an equivalent for bipartite ERGMs

Literature

H1a : Users with the same loan use are likely to have the same credit rating

  • Studied by Serrano-Cinca, Gutiérrez-Nieto, & López-Palacios (2015)

    • Studies determinants of default with P2P lending data from LendingClub

    • Identifies Loan Purpose as significant predictor of default

    • Loans for weddings least risky but small business are riskiest

    • \(\Rightarrow\) Credit riskiness significantly associated with loan purpose

    • \(\Rightarrow\) If loan types vary in riskiness, then users should share creditworthiness

H2a : Pairs of borrowers share same loan type

  • Studied by Liu, Baals, Osterrieder, & Hadji-Misheva (2024)

    • Studies impact of Bondora network centrality on credit risk

    • Edges weighted by Gower distance based on borrower similarity characteristics

    • Analyses degree centrality relative to defaulted vs non-defaulted loans

    • Concludes that degree centrality significant factor in explaining credit risk

    • \(\Rightarrow\) If similarity between borrowers directly explains credit riskiness, then users should cluster around loans sharing similar risk profile (Serrano-Cinca et al., 2015)

H3a : Borrowers do not have a diverse loan use portfolio

  • Studied by Lim, Yeo, Lo, & Tan (2023) and Abbasi, Alam, Brohi, Brohi, & Nasim (2021)

    • General literature review of P2P lending on variety of individuals, including businesses

    • Across Malaysia, most common borrowing is for business financing by SMEs

    • Millennials are the most populous user base of P2P lending platforms

    • Authors argue business loans most popular because traditional commercial loans less accessible and more expensive in Malaysia

    • However, Malaysian and SME context not directly applicable in Bondora’s market

      • Abbasi et al. (2021) studies SME financing in OECD countries between 2011-2018 with P2P borrowing services

      • Authors argue that OECD SMEs rely less on P2P because of greater accessibility of traditional loans

      • \(\Rightarrow\) Traditional European P2P users less likely to demand business loans \(\rightarrow\) more diversity

H4a : Younger borrowers have a different loan use mix

  • Explored by Cooper, Gorbachev, & Luengo‐Prado (2023)

    • Study of credit consumption patterns of younger individuals

    • Authors find that younger individuals have less access to certain categories of traditional credit products and suggests that their credit portfolio structurally different than older borrowers

    • Credit building fundamentally different in U.S., however, attitude towards credit type relevant

    • \(\Rightarrow\) If age is a factor in credit type, then we expect to see a difference in loan use across age

H5a : Borrowers of differing genders have different portfolios of loan use

  • Explored by Aliano, Alnabulsi, Cestari, & Ragni (2023) and Serrano-Cinca et al. (2015)

    • Aliano et al. (2023) study of role of gender and education on probability of default with Bondora’s dataset

    • Authors find a consistent gender effect; women tend to have lower default rates on health, home, and business loans \(\rightarrow\) structural gender differences in borrowing dynamics

      • \(\Rightarrow\) Given that loan purpose is directly related to loan default (Serrano-Cinca et al., 2015), we can expect that the loan use mix has a gender effect

Data Preprocessing

  • Select subset of 112 attributes
c("LoanId", "UserName", "Age", "Gender","Country", "Amount", "Interest","LoanDuration","UseOfLoan", 
  "Education", "MaritalStatus","Rating", "Restructured", "MonthlyPayment")

  • Remove any loans with NA for any of the above features

  • Remove loans where UseOfLoan == -1 to ensure complete data

  • Ensure dataset contains active users: keep UserName with 5+ loans

  • Create incidence matrix to construct partitions \(\mathrm{B}_1\) and \(\mathrm{B}_2\)

  • For ERGM models:

    • Create bipartite network object and add relevant vertex attributes

  • For Linear QAP models:

    • Create adjacency matrices through matrix multiplication: \(\mathbf{X}\cdot\mathbf{X}^{T}\)

    • Ensure no loops can exist by encoding all diagonals as zero

Basic Network Statistics

Study 1 | Preliminary QAP Results

  • Linear QAP Regression used to study how borrowers with the same loan type choice are affected by having the same credit rating

  • The adjacency matrices used are weighted by count to give this study a level of detail we do not have with ERGM

    • Because the weights vary significantly, standardised results are more meaningful to compare

  • Intercept has very significant impact: not all effects can be accounted for and are missing

Variable Type Variable Name
Dependent Variable Weighted Adjacency Matrix of Borrowers with same Loan Type
Explanatory Variable Weighted Adjacency Matrix of Borrowers of the same Credit Rating
Control Variable Weighted Adjacency Matrix of Mean Loan Amount Differences
Control Variable Weighted Adjacency Matrix of Age Differences
Control Variable Binary Adjacency Matrix of Homophily in Gender
Control Variable Weighted Adjacency Matrix of Mean Loan Duration Differences
Control Variable Weighted Adjacency Matrix of Homophily in Loan Restructuring 
qap_m1 <- sna::netlm(y = loan_use_mat,
                     x = list(rating_mat, amt_diffs_mat, age_diffs_mat,
                              gender_mat, loandur_diffs_mat, rest_mat),
                     nullhyp = "qapspp", reps = 2500)

# Standardised QAP Linear Regression
scaled_dep <- scale(loan_use_mat)
scaled_pred <- lapply(pred_vars, scale)

qap_m2 <- sna::netlm(y = scaled_dep,
                     x = scaled_pred,
                     nullhyp = "qapspp", reps = 2500)
              Unstd_Parameters Ustd_t_vals Std_Parameters Std_t_vals
Intercept               6.6104       24.18         0.0050       0.69
Rating                  0.1542        9.55         0.0444       6.01
Loan Amount            -0.0005       -3.67        -0.0266      -3.63
Age                     0.0121        1.30         0.0036       0.49
Gender                  0.1557        0.91         0.0046       0.64
Loan Duration          -0.0063       -0.64        -0.0107      -1.46
Restructured            0.0133        2.38         0.0186       2.51

Study 2 | ERGM | edges

  • Sparse edges but bipartite, so take edges with grain of salt

  • Unconstrained, minimal model

  • GOF acceptable for an initial non-tuned ERGM

formula_base_model <- bondora_net ~ edges
base_ergm <- ergm::ergm(formula_base_model)

===========================
                Model 1    
---------------------------
edges             -1.31 ***
                  (0.07)   
---------------------------
AIC             1287.05    
BIC             1292.17    
Log Likelihood  -642.52    
===========================
*** p < 0.001; ** p < 0.01; * p < 0.05

Study 2 | ERGM | gwb1degree

Modelling Notes

  • GOF indicates that b2degree not modelled well

    • Partition 2 degrees distributed very differently than Partition 1

  • Model 2 converges optimally with MCMC.samplesize = 100000 and MCMC.interval = 1000

  • Greater model stability with MCMC.steplength = 0.25

  • No model constraints

Results

  • Highly positive and significant estimate for gwb1degree which is approximately 0.99 in the odds scale

    • This indicates that borrowers generally have a balanced loan use mix since it is almost equally likely the mix is either highly varied or very specific, relative to a network of random borrowers

  • Excellent MCMC diagnostics

model_1_params <- bondora_net ~ edges + 
  # b1 decay can be very low since 9 b2
  gwb1degree(decay=0.15, fixed=TRUE) 

model_1 <- ergm::ergm(
  model_1_params,
  
  # Max b2 degree is 72, so this constraint is reasonable
  # and helps convergence significantly.
  # Technically in the Bondora population this can be 
  # far higher but we are studying a subsample.
  #constraints = ~ bd(minout = 0, maxout = 80),
  
  control = ergm::control.ergm(
    # Greater burn-in for cleaner result
    MCMC.burnin = 20000,
    # Greater sample size for greater stability
    MCMC.samplesize = 100000,
    seed = 42,
    MCMC.interval = 1000,
    # Only needed for convergence pvals to improve
    MCMLE.maxit = 45,
    # Smaller steps for stability
    MCMLE.steplength = 0.25,
    parallel = n_cores,
    parallel.type = "PSOCK"
  )
)

============================================
                    Model 1      Model 2    
--------------------------------------------
edges                 -1.31 ***    -1.86 ***
                      (0.07)       (0.14)   
gwb1deg.fixed.0.15                  4.53 ***
                                   (1.36)   
--------------------------------------------
AIC                 1287.05      1248.15    
BIC                 1292.17      1258.40    
Log Likelihood      -642.52      -622.07    
============================================
*** p < 0.001; ** p < 0.01; * p < 0.05

Sample statistics summary:

Iterations = 278500:5555500
Thinning interval = 1500 
Number of chains = 9 
Sample size per chain = 3519 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

                       Mean     SD Naive SE Time-series SE
edges              -1.63730 10.142 0.056991       0.056430
gwb1deg.fixed.0.15  0.09889  1.041 0.005849       0.005796

2. Quantiles for each variable:

                      2.5%     25%    50%    75%  97.5%
edges              -21.000 -9.0000 -2.000 5.0000 19.000
gwb1deg.fixed.0.15  -2.169 -0.5529  0.166 0.8252  1.943


Are sample statistics significantly different from observed?
                    edges gwb1deg.fixed.0.15  (Omni)
diff.       -1.637302e+00       9.888600e-02      NA
test stat.  -2.905064e+01       1.707669e+01 3907.74
P-val.      1.510385e-185       2.213288e-65    0.00

Sample statistics cross-correlations:
                       edges gwb1deg.fixed.0.15
edges              1.0000000          0.7307488
gwb1deg.fixed.0.15 0.7307488          1.0000000

Sample statistics auto-correlation:
Chain 1 
                edges gwb1deg.fixed.0.15
Lag 0     1.000000000        1.000000000
Lag 1500  0.012838107        0.015140352
Lag 3000  0.006146927        0.005668116
Lag 4500 -0.021018674       -0.005528812
Lag 6000  0.013435694        0.010964681
Lag 7500  0.018525684        0.028763557
Chain 2 
                edges gwb1deg.fixed.0.15
Lag 0     1.000000000        1.000000000
Lag 1500  0.001043010       -0.004401243
Lag 3000 -0.002825356       -0.011152768
Lag 4500 -0.008965341       -0.021251946
Lag 6000  0.004859929       -0.009722392
Lag 7500 -0.012258256       -0.001149459
Chain 3 
                 edges gwb1deg.fixed.0.15
Lag 0     1.0000000000        1.000000000
Lag 1500  0.0033366717       -0.009107698
Lag 3000 -0.0125214237       -0.020143814
Lag 4500  0.0138190231        0.027417159
Lag 6000 -0.0009386103       -0.020902170
Lag 7500  0.0189907172        0.004616177
Chain 4 
                edges gwb1deg.fixed.0.15
Lag 0     1.000000000        1.000000000
Lag 1500 -0.003664673        0.014542298
Lag 3000 -0.005028443        0.014937576
Lag 4500 -0.017851999       -0.008323828
Lag 6000  0.003740486       -0.020081406
Lag 7500 -0.010728196       -0.008067818
Chain 5 
                 edges gwb1deg.fixed.0.15
Lag 0     1.0000000000        1.000000000
Lag 1500 -0.0115820184        0.015833319
Lag 3000  0.0002050104       -0.011459640
Lag 4500 -0.0066442933        0.010544615
Lag 6000 -0.0216394887       -0.015178084
Lag 7500 -0.0115478036       -0.002827164
Chain 6 
                edges gwb1deg.fixed.0.15
Lag 0     1.000000000       1.0000000000
Lag 1500  0.003780637      -0.0282198840
Lag 3000  0.019772542       0.0351128668
Lag 4500 -0.012204203       0.0002017706
Lag 6000  0.016252017      -0.0035913917
Lag 7500  0.019921304       0.0221747935
Chain 7 
                edges gwb1deg.fixed.0.15
Lag 0     1.000000000        1.000000000
Lag 1500 -0.012713966        0.012824455
Lag 3000  0.014580153        0.002109272
Lag 4500 -0.006131341       -0.004317167
Lag 6000 -0.016162120       -0.003013537
Lag 7500  0.019596084        0.039034221
Chain 8 
                edges gwb1deg.fixed.0.15
Lag 0     1.000000000        1.000000000
Lag 1500 -0.008425015        0.011839598
Lag 3000 -0.023633086        0.005255397
Lag 4500 -0.046501743       -0.054864848
Lag 6000 -0.003264238       -0.005825029
Lag 7500 -0.009768808        0.013437188
Chain 9 
                edges gwb1deg.fixed.0.15
Lag 0     1.000000000        1.000000000
Lag 1500 -0.008782055       -0.003747304
Lag 3000 -0.041016476       -0.051435341
Lag 4500  0.020263431        0.008928846
Lag 6000 -0.020265219       -0.020456047
Lag 7500  0.034069472        0.012796840

Sample statistics burn-in diagnostic (Geweke):
Chain 1 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15 
         1.0840297          0.1105535 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15 
         0.2783516          0.9119705 
Joint P-value (lower = worse):  0.3564513 
Chain 2 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15 
          1.065432           1.128339 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15 
         0.2866804          0.2591770 
Joint P-value (lower = worse):  0.482874 
Chain 3 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15 
         0.4451861         -0.1452447 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15 
         0.6561853          0.8845177 
Joint P-value (lower = worse):  0.6811179 
Chain 4 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15 
          1.942605           1.747066 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15 
        0.05206387         0.08062581 
Joint P-value (lower = worse):  0.1505848 
Chain 5 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15 
         -1.902738          -1.275174 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15 
        0.05707469         0.20224780 
Joint P-value (lower = worse):  0.1984048 
Chain 6 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15 
          2.117553           1.094498 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15 
        0.03421293         0.27373644 
Joint P-value (lower = worse):  0.0736976 
Chain 7 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15 
         0.1724423         -0.3260063 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15 
         0.8630898          0.7444196 
Joint P-value (lower = worse):  0.7710591 
Chain 8 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15 
          1.003970           1.216699 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15 
         0.3153929          0.2237187 
Joint P-value (lower = worse):  0.4791229 
Chain 9 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15 
        -0.7107651         -0.7848914 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15 
         0.4772298          0.4325172 
Joint P-value (lower = worse):  0.719484 

Note: To save space, only one in every 3 iterations of the MCMC sample
  used for estimation was stored for diagnostics. Sample size per chain
  was originally around 10557 with thinning interval 500.

Note: MCMC diagnostics shown here are from the last round of
  simulation, prior to computation of final parameter estimates.
  Because the final estimates are refinements of those used for this
  simulation run, these diagnostics may understate model performance.
  To directly assess the performance of the final model on in-model
  statistics, please use the GOF command: gof(ergmFitObject,
  GOF=~model).

Study 2 | ERGM | gwb1dsp

Modelling Notes

  • Model constrained by minimum of zero outdegrees and maximum of 80

    • Smaller simulation space reduces the noisiness of the MCMC mixing process significantly

    • Maximum b2degree=72 so 80 may be reasonable, but we understand the conceptual implications of this

  • gwb1dsp decay was set to 0.5 because there are far too many dyad-wise partners otherwise

    • Cannot be set too high otherwise too many relevant dyad-wise partners are lost

Results

  • gwb1dsp estimate is highly significant and approximately 0.51 in the odds scale

    • This indicates that sets of two borrowers are less likely to share the same loan type compared to a random case where borrowers select their loan type randomly

  • MCMC diagnostics indicate good mixing, though gwb1dsp traces are right-skewed slightly

  • GOF acceptable, however, modelling b2degree is difficult

  • AIC/BIC highly negative as a result \(\rightarrow\) Model 3 theoretically better than Model 2 given MCMC + GOF

    • Extremely low Standard Error for gwb1dsp
model_2_params <- bondora_net ~ edges + 
  # low decay important because there is high clustering around low degrees
  gwb1degree(decay=0.15, fixed=TRUE) + 
  # decay should be higher due to wider variation in degree but 
  # too high of degree makes the traces concentrated around the tails.
  gwb1dsp(decay=0.5, fixed=TRUE)      
  
model_2 <- ergm::ergm(
  model_2_params,
  
  # Max b2 degree is 72, so this constraint is reasonable
  # and helps convergence significantly.
  # Technically in the Bondora population this can be 
  # far higher but we are studying a subsample.
  constraints = ~ bd(minout = 0, maxout = 80),
  
  control = ergm::control.ergm(
    # Greater burn-in for cleaner result
    MCMC.burnin = 20000,
    # Greater sample size for greater stability
    MCMC.samplesize = 100000,
    seed = 42,
    MCMC.interval = 1000,
    # Only needed for convergence pvals to improve
    MCMLE.maxit = 45,
    # Smaller steps for stability
    MCMLE.steplength = 0.25,
    parallel = n_cores,
    parallel.type = "PSOCK"
    )
  )

=========================================================
                    Model 1      Model 2      Model 3    
---------------------------------------------------------
edges                 -1.31 ***    -1.86 ***    -2.76 ***
                      (0.07)       (0.14)       (0.13)   
gwb1deg.fixed.0.15                  4.53 ***     3.01 ** 
                                   (1.36)       (1.04)   
gwb1dsp.fixed.0.5                                0.04 ***
                                                (0.00)   
---------------------------------------------------------
AIC                 1287.05      1248.15      -581.93    
BIC                 1292.17      1258.40      -566.55    
Log Likelihood      -642.52      -622.07       293.96    
=========================================================
*** p < 0.001; ** p < 0.01; * p < 0.05

Sample statistics summary:

Iterations = 51460:1028580
Thinning interval = 248 
Number of chains = 9 
Sample size per chain = 3941 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

                       Mean      SD Naive SE Time-series SE
edges               -1.5670  12.209 0.064829        0.13831
gwb1deg.fixed.0.15  -0.3265   1.496 0.007944        0.01713
gwb1dsp.fixed.0.5  -76.7347 431.450 2.290899       16.47459

2. Quantiles for each variable:

                       2.5%      25%       50%     75%    97.5%
edges               -25.000  -10.000   -2.0000   7.000   23.000
gwb1deg.fixed.0.15   -3.551   -1.258   -0.2002   0.724    2.261
gwb1dsp.fixed.0.5  -730.149 -361.725 -151.1852 115.166 1051.838


Are sample statistics significantly different from observed?
                   edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5       (Omni)
diff.      -1.567002e+00      -3.264965e-01     -7.673467e+01           NA
test stat. -1.140165e+01      -1.932060e+01     -5.537013e+00 4.861427e+02
P-val.      4.102551e-30       3.604419e-83      3.076736e-08 2.406545e-99

Sample statistics cross-correlations:
                       edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
edges              1.0000000          0.7445521         0.6367880
gwb1deg.fixed.0.15 0.7445521          1.0000000         0.5618037
gwb1dsp.fixed.0.5  0.6367880          0.5618037         1.0000000

Sample statistics auto-correlation:
Chain 1 
              edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.00000000         1.00000000         1.0000000
Lag 248  0.56295598         0.50602290         0.7040632
Lag 496  0.32046249         0.25174571         0.5499268
Lag 744  0.17734685         0.11327805         0.4475817
Lag 992  0.10858322         0.06271803         0.3839918
Lag 1240 0.05465916         0.03923626         0.3388017
Chain 2 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 248  0.5980003          0.5434762         0.8782729
Lag 496  0.3694831          0.3239478         0.8090821
Lag 744  0.2329017          0.2166474         0.7652058
Lag 992  0.1709862          0.1601746         0.7379846
Lag 1240 0.1417389          0.1334169         0.7246901
Chain 3 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 248  0.6089901          0.5423317         0.7843989
Lag 496  0.4027002          0.3401747         0.6682450
Lag 744  0.2859170          0.2385278         0.5960251
Lag 992  0.2040205          0.1736689         0.5335211
Lag 1240 0.1546055          0.1387653         0.4967728
Chain 4 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 248  0.6021103          0.5133558         0.8395708
Lag 496  0.3561906          0.2774625         0.7463385
Lag 744  0.2150795          0.1828983         0.6848262
Lag 992  0.1244300          0.1246336         0.6396067
Lag 1240 0.0764468          0.0868754         0.6108981
Chain 5 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 248  0.6291523          0.5520590         0.8753196
Lag 496  0.4029451          0.3294004         0.8023333
Lag 744  0.2678178          0.1986287         0.7546725
Lag 992  0.2009926          0.1651171         0.7230272
Lag 1240 0.1403551          0.1336568         0.6976788
Chain 6 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 248  0.5878859          0.5269955         0.8353683
Lag 496  0.3719123          0.3097583         0.7487089
Lag 744  0.2519750          0.2184473         0.6978643
Lag 992  0.1976614          0.1554344         0.6692603
Lag 1240 0.1514289          0.1303012         0.6455738
Chain 7 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 248  0.5929592          0.5174850         0.8221950
Lag 496  0.3783554          0.2875448         0.7274247
Lag 744  0.2403588          0.1747668         0.6679365
Lag 992  0.1501032          0.1164512         0.6154744
Lag 1240 0.1328045          0.1189623         0.5948866
Chain 8 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 248  0.5920054          0.5670022         0.8084281
Lag 496  0.3872846          0.3694622         0.7149825
Lag 744  0.2576075          0.2516709         0.6594858
Lag 992  0.1773724          0.1748777         0.6150392
Lag 1240 0.1153876          0.1410917         0.5843777
Chain 9 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 248  0.6108218          0.5503831         0.7964875
Lag 496  0.3905985          0.3251249         0.6886352
Lag 744  0.2491485          0.2314349         0.6148121
Lag 992  0.1683030          0.1559633         0.5690688
Lag 1240 0.1148987          0.1232386         0.5283541

Sample statistics burn-in diagnostic (Geweke):
Chain 1 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
        0.76194086         0.95737330         0.01953038 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
         0.4460953          0.3383789          0.9844180 
Joint P-value (lower = worse):  0.7681905 
Chain 2 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
          2.092323           2.097187           1.115353 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
        0.03640963         0.03597702         0.26469938 
Joint P-value (lower = worse):  0.2648371 
Chain 3 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
        -0.5730499         -0.2956200          0.2532658 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
         0.5666109          0.7675203          0.8000629 
Joint P-value (lower = worse):  0.7882049 
Chain 4 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
         0.1700013          0.2818040         -0.2642180 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
         0.8650091          0.7780938          0.7916119 
Joint P-value (lower = worse):  0.9048597 
Chain 5 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
         -0.915613          -0.207188          -2.466979 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
        0.35986993         0.83586305         0.01362585 
Joint P-value (lower = worse):  0.05846161 
Chain 6 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
         -1.206604          -2.108245          -1.107644 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
        0.22758458         0.03500977         0.26801555 
Joint P-value (lower = worse):  0.8003396 
Chain 7 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
         0.4438510          1.0863773          0.4456894 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
         0.6571503          0.2773121          0.6558216 
Joint P-value (lower = worse):  0.7302439 
Chain 8 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
        -0.4770260         -0.5568458         -1.7745149 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
        0.63334363         0.57763278         0.07597802 
Joint P-value (lower = worse):  0.8034316 
Chain 9 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
         0.2994310          0.4308009          1.1339605 

Individual P-values (lower = worse):
             edges gwb1deg.fixed.0.15  gwb1dsp.fixed.0.5 
         0.7646112          0.6666131          0.2568111 
Joint P-value (lower = worse):  0.5345944 

Note: To save space, only one in every 8 iterations of the MCMC sample
  used for estimation was stored for diagnostics. Sample size per chain
  was originally around 31528 with thinning interval 31.

Note: MCMC diagnostics shown here are from the last round of
  simulation, prior to computation of final parameter estimates.
  Because the final estimates are refinements of those used for this
  simulation run, these diagnostics may understate model performance.
  To directly assess the performance of the final model on in-model
  statistics, please use the GOF command: gof(ergmFitObject,
  GOF=~model).

Study 2 | ERGM | b1nodematch("gender")

Modelling Notes

  • Since b1nodematch is an exogenous dyad-independent term, the parameters related to convergence were mostly unaffected

Results

  • b1nodematch is used as a proxy for nodemix because it is unavailable for bipartite networks

  • b1nodematch significant at 5% and converted to the odds scale, estimate is 0.51

    • This indicates that being of the same gender decreases the diversity of the borrower’s requested loan types, compared to being of the other gender, all else equal

  • The inclusion of the b1nodematch term increased the strength of the gwb1degree term, but decreased gwb1dsp

  • The term decreased the significance of the gwb1dsp term, suggesting it may be less statistically relevant under this modelling context

  • MCMC diagnostics show that b1nodematch worsens convergence and skews the traces of the other terms, however, it is acceptable

  • GOF is similar to previous cases, however, the fit for b1degree is slightly less optimal

  • AIC suggests that this model offers less explanatory potential than the previous, however, BIC indicates the converse. Since the difference is small, this model is not seen as significantly better formulated

model_3_params <- bondora_net ~ edges + 
  # low decay important because there is high clustering around low degrees
  gwb1degree(decay=0.15, fixed=TRUE) + 
  # decay should be higher due to wider variation in degree but 
  # too high of degree makes the traces concentrated around the tails.
  gwb1dsp(decay=0.5, fixed=TRUE) +
  # See differences across genders (implicitly, since b1nodemix unavailable)
  b1nodematch("b1_gender", diff=FALSE)

model_3 <- ergm::ergm(
  model_3_params,
  
  # Max b2 degree is 72, so this constraint is reasonable
  # and helps convergence significantly.
  # Technically in the Bondora population this can be 
  # far higher but we are studying a subsample.
  constraints = ~ bd(minout = 0, maxout = 80),
  
  control = ergm::control.ergm(
    # Greater burn-in for cleaner result
    MCMC.burnin = 20000,
    # Greater sample size for greater stability
    MCMC.samplesize = 100000,
    seed = 42,
    MCMC.interval = 1000,
    # Only needed for convergence pvals to improve
    MCMLE.maxit = 45,
    # Smaller steps for stability
    MCMLE.steplength = 0.25,
    parallel = n_cores,
    parallel.type = "PSOCK"
  )
)

=========================================================================
                       Model 1      Model 2      Model 3      Model 4    
-------------------------------------------------------------------------
edges                    -1.31 ***    -1.86 ***    -2.76 ***    -2.86 ***
                         (0.07)       (0.14)       (0.13)       (0.13)   
gwb1deg.fixed.0.15                     4.53 ***     3.01 **      3.46 ** 
                                      (1.36)       (1.04)       (1.15)   
gwb1dsp.fixed.0.5                                   0.04 ***     0.02 *  
                                                   (0.00)       (0.01)   
b1nodematch.b1_gender                                            0.04 ** 
                                                                (0.01)   
-------------------------------------------------------------------------
AIC                    1287.05      1248.15      -581.93      -583.90    
BIC                    1292.17      1258.40      -566.55      -563.40    
Log Likelihood         -642.52      -622.07       293.96       295.95    
=========================================================================
*** p < 0.001; ** p < 0.01; * p < 0.05

Sample statistics summary:

Iterations = 52516:218968
Thinning interval = 44 
Number of chains = 9 
Sample size per chain = 3784 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

                          Mean      SD Naive SE Time-series SE
edges                  0.78888  13.002 0.070455        0.34132
gwb1deg.fixed.0.15     0.04787   1.407 0.007623        0.03328
gwb1dsp.fixed.0.5      5.86648 533.968 2.893467       32.13073
b1nodematch.b1_gender 16.04857 339.461 1.839472       19.84893

2. Quantiles for each variable:

                          2.5%       25%       50%     75%    97.5%
edges                  -24.000   -8.0000    1.0000  10.000   26.000
gwb1deg.fixed.0.15      -3.043   -0.8263    0.1454   1.032    2.496
gwb1dsp.fixed.0.5     -789.747 -354.2152 -113.8707 251.509 1270.056
b1nodematch.b1_gender -488.000 -221.0000  -60.0000 189.000  817.000


Are sample statistics significantly different from observed?
                edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
diff.      0.78887714         0.04787403         5.8664830
test stat. 2.30633565         1.43899395         0.2484816
P-val.     0.02109188         0.15015224         0.8037618
           b1nodematch.b1_gender       (Omni)
diff.                 16.0485671           NA
test stat.             1.0399635 17.588525639
P-val.                 0.2983568  0.001667632

Sample statistics cross-correlations:
                          edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
edges                 1.0000000          0.7732521         0.6384334
gwb1deg.fixed.0.15    0.7732521          1.0000000         0.5798158
gwb1dsp.fixed.0.5     0.6384334          0.5798158         1.0000000
b1nodematch.b1_gender 0.6631398          0.5799913         0.9763882
                      b1nodematch.b1_gender
edges                             0.6631398
gwb1deg.fixed.0.15                0.5799913
gwb1dsp.fixed.0.5                 0.9763882
b1nodematch.b1_gender             1.0000000

Sample statistics auto-correlation:
Chain 1 
            edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5 b1nodematch.b1_gender
Lag 0   1.0000000          1.0000000         1.0000000             1.0000000
Lag 44  0.9319187          0.9024248         0.9804028             0.9780253
Lag 88  0.8701917          0.8184135         0.9630878             0.9578511
Lag 132 0.8132989          0.7485662         0.9468151             0.9391379
Lag 176 0.7596956          0.6949520         0.9309139             0.9207637
Lag 220 0.7103323          0.6473736         0.9163540             0.9038319
Chain 2 
            edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5 b1nodematch.b1_gender
Lag 0   1.0000000          1.0000000         1.0000000             1.0000000
Lag 44  0.9223415          0.8928383         0.9795651             0.9785882
Lag 88  0.8526729          0.8007243         0.9620724             0.9600706
Lag 132 0.7874353          0.7170862         0.9462147             0.9429185
Lag 176 0.7283505          0.6456673         0.9315790             0.9272254
Lag 220 0.6753385          0.5848759         0.9179151             0.9126281
Chain 3 
            edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5 b1nodematch.b1_gender
Lag 0   1.0000000          1.0000000         1.0000000             1.0000000
Lag 44  0.9266632          0.9022882         0.9741064             0.9739837
Lag 88  0.8624248          0.8175749         0.9525465             0.9524120
Lag 132 0.8025855          0.7433882         0.9326125             0.9319238
Lag 176 0.7444362          0.6785781         0.9145351             0.9130541
Lag 220 0.6869249          0.6169745         0.8968737             0.8944530
Chain 4 
            edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5 b1nodematch.b1_gender
Lag 0   1.0000000          1.0000000         1.0000000             1.0000000
Lag 44  0.9111857          0.8808525         0.9246058             0.9360860
Lag 88  0.8276554          0.7721150         0.8558146             0.8762054
Lag 132 0.7527559          0.6765096         0.7974222             0.8242702
Lag 176 0.6878365          0.5919269         0.7464735             0.7789254
Lag 220 0.6298961          0.5189842         0.7000681             0.7378573
Chain 5 
            edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5 b1nodematch.b1_gender
Lag 0   1.0000000          1.0000000         1.0000000             1.0000000
Lag 44  0.9091602          0.8846306         0.9145542             0.9226400
Lag 88  0.8241148          0.7864740         0.8370257             0.8510909
Lag 132 0.7501656          0.7052962         0.7684393             0.7886067
Lag 176 0.6850116          0.6360706         0.7095837             0.7344858
Lag 220 0.6268410          0.5758196         0.6555811             0.6854856
Chain 6 
            edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5 b1nodematch.b1_gender
Lag 0   1.0000000          1.0000000         1.0000000             1.0000000
Lag 44  0.9240690          0.8998647         0.9725435             0.9736895
Lag 88  0.8567303          0.8162380         0.9481483             0.9503893
Lag 132 0.7935653          0.7455213         0.9250406             0.9286096
Lag 176 0.7388855          0.6812359         0.9047833             0.9098259
Lag 220 0.6901205          0.6212459         0.8870261             0.8931279
Chain 7 
            edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5 b1nodematch.b1_gender
Lag 0   1.0000000          1.0000000         1.0000000             1.0000000
Lag 44  0.9097246          0.8715946         0.9634483             0.9629883
Lag 88  0.8283612          0.7716534         0.9320415             0.9304241
Lag 132 0.7523051          0.6848507         0.9028355             0.9001209
Lag 176 0.6867768          0.6163924         0.8777351             0.8735004
Lag 220 0.6295303          0.5602284         0.8541161             0.8481965
Chain 8 
            edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5 b1nodematch.b1_gender
Lag 0   1.0000000          1.0000000         1.0000000             1.0000000
Lag 44  0.9256862          0.8936635         0.9480094             0.9456491
Lag 88  0.8631946          0.8052372         0.9055970             0.8997843
Lag 132 0.8040650          0.7286670         0.8652400             0.8552045
Lag 176 0.7507139          0.6640299         0.8304690             0.8167076
Lag 220 0.6901306          0.5992848         0.7913364             0.7737518
Chain 9 
            edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5 b1nodematch.b1_gender
Lag 0   1.0000000          1.0000000         1.0000000             1.0000000
Lag 44  0.9098865          0.8698457         0.9538337             0.9573081
Lag 88  0.8281999          0.7611854         0.9145370             0.9202462
Lag 132 0.7593651          0.6628772         0.8839200             0.8900558
Lag 176 0.6950981          0.5749144         0.8555155             0.8621501
Lag 220 0.6379381          0.5008714         0.8310153             0.8374309

Sample statistics burn-in diagnostic (Geweke):
Chain 1 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
            -1.483805             -2.349380             -2.071158 
b1nodematch.b1_gender 
            -2.649035 

Individual P-values (lower = worse):
                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
          0.137860549           0.018804714           0.038344055 
b1nodematch.b1_gender 
          0.008072203 
Joint P-value (lower = worse):  0.01677586 
Chain 2 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
            -1.534365             -1.457523             -2.261492 
b1nodematch.b1_gender 
            -1.584850 

Individual P-values (lower = worse):
                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
            0.1249399             0.1449720             0.0237288 
b1nodematch.b1_gender 
            0.1130004 
Joint P-value (lower = worse):  0.1609923 
Chain 3 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
           -0.6334113            -0.4737616            -0.7990936 
b1nodematch.b1_gender 
           -1.0658562 

Individual P-values (lower = worse):
                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
            0.5264651             0.6356699             0.4242361 
b1nodematch.b1_gender 
            0.2864886 
Joint P-value (lower = worse):  0.8017936 
Chain 4 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
           -1.3674791            -0.7764809            -0.8251074 
b1nodematch.b1_gender 
           -0.6002265 

Individual P-values (lower = worse):
                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
            0.1714752             0.4374651             0.4093106 
b1nodematch.b1_gender 
            0.5483553 
Joint P-value (lower = worse):  0.8235535 
Chain 5 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
           -0.3700723             0.1370897             0.2044858 
b1nodematch.b1_gender 
            1.1249478 

Individual P-values (lower = worse):
                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
            0.7113286             0.8909599             0.8379739 
b1nodematch.b1_gender 
            0.2606112 
Joint P-value (lower = worse):  0.2470521 
Chain 6 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
             5.722266              7.098553             12.323519 
b1nodematch.b1_gender 
            10.604526 

Individual P-values (lower = worse):
                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
         1.051125e-08          1.260695e-12          6.767505e-35 
b1nodematch.b1_gender 
         2.839007e-26 
Joint P-value (lower = worse):  4.014915e-09 
Chain 7 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
            1.8947553             0.3110727            -1.2974889 
b1nodematch.b1_gender 
           -1.1566834 

Individual P-values (lower = worse):
                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
           0.05812482            0.75574533            0.19446301 
b1nodematch.b1_gender 
           0.24740174 
Joint P-value (lower = worse):  0.01484358 
Chain 8 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
          -1.97199985           -2.42485587            0.02463288 
b1nodematch.b1_gender 
          -0.15983537 

Individual P-values (lower = worse):
                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
           0.04860962            0.01531447            0.98034779 
b1nodematch.b1_gender 
           0.87301076 
Joint P-value (lower = worse):  0.3385534 
Chain 9 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
            -1.689707             -1.756122             -1.147265 
b1nodematch.b1_gender 
            -2.287856 

Individual P-values (lower = worse):
                edges    gwb1deg.fixed.0.15     gwb1dsp.fixed.0.5 
           0.09108404            0.07906766            0.25127199 
b1nodematch.b1_gender 
           0.02214589 
Joint P-value (lower = worse):  0.3793556 

Note: To save space, only one in every 11 iterations of the MCMC sample
  used for estimation was stored for diagnostics. Sample size per chain
  was originally around 41624 with thinning interval 4.

Note: MCMC diagnostics shown here are from the last round of
  simulation, prior to computation of final parameter estimates.
  Because the final estimates are refinements of those used for this
  simulation run, these diagnostics may understate model performance.
  To directly assess the performance of the final model on in-model
  statistics, please use the GOF command: gof(ergmFitObject,
  GOF=~model).

Study 2 | ERGM | b1cov("age")

Modelling Notes

  • Parameters related to convergence were unchanged since we have added an exogenous, dyad-independent term

Results

  • The b1cov term is statistically insignificant and the magnitude is zero, indicating that it is irrelevant to the explanatory power of this model

  • The inclusion of the b1cov term appears to stabilise the MCMC mixing process, improving the trace distributions

  • The GOF is not significantly different than previous iterations

  • BIC, which penalises model complexity more than AIC, is notably better than the previous models \(\Rightarrow\) this model offers better statistical relevance than the previous models, given our subsample

model_4_params <- bondora_net ~ edges + 
  # low decay important because there is high clustering around low degrees
  gwb1degree(decay=0.15, fixed=TRUE) + 
  # decay should be higher due to wider variation in degree but 
  # too high of degree makes the traces concentrated around the tails.
  gwb1dsp(decay=0.5, fixed=TRUE) +
  # See differences across genders (implicitly, since b1nodemix unavailable)
  b1nodematch("b1_gender", diff=TRUE) +
  # See if higher ages make a difference
  b1cov("b1_age")

model_4 <- ergm::ergm(
  model_4_params,
  
  # Max b2 degree is 72, so this constraint is reasonable
  # and helps convergence significantly.
  # Technically in the Bondora population this can be 
  # far higher but we are studying a subsample.
  constraints = ~ bd(minout = 0, maxout = 80),
  
  control = ergm::control.ergm(
    # Greater burn-in for cleaner result
    MCMC.burnin = 20000,
    # Greater sample size for greater stability
    MCMC.samplesize = 100000,
    seed = 42,
    MCMC.interval = 1000,
    # Only needed for convergence pvals to improve
    MCMLE.maxit = 45,
    # Smaller steps for stability
    MCMLE.steplength = 0.25,
    parallel = n_cores,
    parallel.type = "PSOCK"
  )
)

=============================================================================================
                              Model 1      Model 2      Model 3      Model 4      Model 5    
---------------------------------------------------------------------------------------------
edges                           -1.31 ***    -1.86 ***    -2.76 ***    -2.86 ***    -2.92 ***
                                (0.07)       (0.14)       (0.13)       (0.13)       (0.39)   
gwb1deg.fixed.0.15                            4.53 ***     3.01 **      3.46 **      3.48 ** 
                                             (1.36)       (1.04)       (1.15)       (1.13)   
gwb1dsp.fixed.0.5                                          0.04 ***     0.02 *       0.02 *  
                                                          (0.00)       (0.01)       (0.01)   
b1nodematch.b1_gender                                                   0.04 **              
                                                                       (0.01)                
b1nodematch.b1_gender.female                                                         0.04 ** 
                                                                                    (0.01)   
b1nodematch.b1_gender.male                                                           0.04 ** 
                                                                                    (0.01)   
b1cov.b1_age                                                                         0.00    
                                                                                    (0.01)   
---------------------------------------------------------------------------------------------
AIC                           1287.05      1248.15      -581.93      -583.90      -580.00    
BIC                           1292.17      1258.40      -566.55      -563.40      -549.25    
Log Likelihood                -642.52      -622.07       293.96       295.95       296.00    
=============================================================================================
*** p < 0.001; ** p < 0.01; * p < 0.05

Sample statistics summary:

Iterations = 53936:1078192
Thinning interval = 256 
Number of chains = 9 
Sample size per chain = 4002 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

                                 Mean      SD Naive SE Time-series SE
edges                        -0.24180  13.290 0.070027        0.20034
gwb1deg.fixed.0.15           -0.04666   1.449 0.007633        0.02313
gwb1dsp.fixed.0.5            11.33742 539.848 2.844535       21.81212
b1nodematch.b1_gender.female  3.85702 243.194 1.281427        8.35787
b1nodematch.b1_gender.male    7.64323 254.272 1.339797        8.29373
b1cov.b1_age                 -6.80310 520.786 2.744096        7.48004

2. Quantiles for each variable:

                                  2.5%       25%        50%      75%    97.5%
edges                          -26.000   -9.0000   -1.00000   9.0000   26.000
gwb1deg.fixed.0.15              -3.188   -0.9539    0.05628   0.9735    2.477
gwb1dsp.fixed.0.5             -809.739 -372.2038 -108.20359 318.1234 1239.381
b1nodematch.b1_gender.female  -419.000 -170.0000  -14.00000 157.0000  525.000
b1nodematch.b1_gender.male    -420.000 -176.0000  -20.00000 166.0000  565.000
b1cov.b1_age                 -1001.000 -367.0000  -16.00000 338.0000 1042.000


Are sample statistics significantly different from observed?
                edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
diff.      -0.2417958        -0.04665638        11.3374239
test stat. -1.2438557        -2.16150682         0.5372249
P-val.      0.2135527         0.03065621         0.5911122
           b1nodematch.b1_gender.female b1nodematch.b1_gender.male b1cov.b1_age
diff.                         3.8570159                  7.6432339   -6.8030985
test stat.                    0.4689786                  0.9311016   -0.9307789
P-val.                        0.6390849                  0.3518010    0.3519680
                 (Omni)
diff.                NA
test stat. 4.283009e+01
P-val.     1.444492e-07

Sample statistics cross-correlations:
                                 edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
edges                        1.0000000          0.7810322         0.6543415
gwb1deg.fixed.0.15           0.7810322          1.0000000         0.5947371
gwb1dsp.fixed.0.5            0.6543415          0.5947371         1.0000000
b1nodematch.b1_gender.female 0.4511137          0.3907929         0.6561411
b1nodematch.b1_gender.male   0.4936680          0.4391739         0.7047266
b1cov.b1_age                 0.9790405          0.7635908         0.6418934
                             b1nodematch.b1_gender.female
edges                                          0.45111369
gwb1deg.fixed.0.15                             0.39079288
gwb1dsp.fixed.0.5                              0.65614107
b1nodematch.b1_gender.female                   1.00000000
b1nodematch.b1_gender.male                    -0.02640127
b1cov.b1_age                                   0.45496460
                             b1nodematch.b1_gender.male b1cov.b1_age
edges                                        0.49366796    0.9790405
gwb1deg.fixed.0.15                           0.43917388    0.7635908
gwb1dsp.fixed.0.5                            0.70472656    0.6418934
b1nodematch.b1_gender.female                -0.02640127    0.4549646
b1nodematch.b1_gender.male                   1.00000000    0.4721943
b1cov.b1_age                                 0.47219434    1.0000000

Sample statistics auto-correlation:
Chain 1 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 256  0.6453460          0.5527544         0.8521714
Lag 512  0.4406780          0.3467446         0.7601589
Lag 768  0.3208653          0.2481458         0.6963447
Lag 1024 0.2408560          0.1980446         0.6502608
Lag 1280 0.1772788          0.1608781         0.6139355
         b1nodematch.b1_gender.female b1nodematch.b1_gender.male b1cov.b1_age
Lag 0                       1.0000000                  1.0000000    1.0000000
Lag 256                     0.8735722                  0.8651314    0.6401510
Lag 512                     0.7907868                  0.7744271    0.4343235
Lag 768                     0.7330084                  0.7113727    0.3156968
Lag 1024                    0.6876556                  0.6624295    0.2317767
Lag 1280                    0.6465640                  0.6279112    0.1658382
Chain 2 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 256  0.6320401          0.5597469         0.8641657
Lag 512  0.4238484          0.3599751         0.7922162
Lag 768  0.2950973          0.2633642         0.7398011
Lag 1024 0.2257244          0.2006848         0.7088217
Lag 1280 0.1740757          0.1645989         0.6833862
         b1nodematch.b1_gender.female b1nodematch.b1_gender.male b1cov.b1_age
Lag 0                       1.0000000                  1.0000000    1.0000000
Lag 256                     0.8777747                  0.8687046    0.6206424
Lag 512                     0.7962103                  0.7895781    0.4135612
Lag 768                     0.7366577                  0.7284120    0.2856665
Lag 1024                    0.6929756                  0.6873869    0.2166030
Lag 1280                    0.6537940                  0.6515058    0.1724934
Chain 3 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 256  0.6459153          0.5673497         0.8500984
Lag 512  0.4411069          0.3730364         0.7557321
Lag 768  0.3204534          0.2629693         0.6884492
Lag 1024 0.2371480          0.2143969         0.6447021
Lag 1280 0.1768998          0.1810011         0.6102217
         b1nodematch.b1_gender.female b1nodematch.b1_gender.male b1cov.b1_age
Lag 0                       1.0000000                  1.0000000    1.0000000
Lag 256                     0.8714958                  0.8735923    0.6408687
Lag 512                     0.7844975                  0.7904100    0.4309246
Lag 768                     0.7199218                  0.7352873    0.3074769
Lag 1024                    0.6749545                  0.6922785    0.2304608
Lag 1280                    0.6377457                  0.6621571    0.1714990
Chain 4 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 256  0.6505369          0.5671874         0.8837727
Lag 512  0.4354328          0.3561811         0.8078543
Lag 768  0.3147191          0.2527893         0.7648243
Lag 1024 0.2481485          0.2034781         0.7332508
Lag 1280 0.2107569          0.1836693         0.7098417
         b1nodematch.b1_gender.female b1nodematch.b1_gender.male b1cov.b1_age
Lag 0                       1.0000000                  1.0000000    1.0000000
Lag 256                     0.8881558                  0.8607624    0.6501827
Lag 512                     0.8147356                  0.7649229    0.4342458
Lag 768                     0.7630334                  0.7057188    0.3137798
Lag 1024                    0.7215932                  0.6643145    0.2457666
Lag 1280                    0.6855579                  0.6352151    0.2072487
Chain 5 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 256  0.6702681          0.5845211         0.9091538
Lag 512  0.4774371          0.4069621         0.8509961
Lag 768  0.3365553          0.2830732         0.8071310
Lag 1024 0.2637747          0.2219023         0.7783353
Lag 1280 0.2254330          0.1813640         0.7559684
         b1nodematch.b1_gender.female b1nodematch.b1_gender.male b1cov.b1_age
Lag 0                       1.0000000                  1.0000000    1.0000000
Lag 256                     0.8810276                  0.8922021    0.6592930
Lag 512                     0.8053990                  0.8183612    0.4666235
Lag 768                     0.7478913                  0.7574796    0.3339456
Lag 1024                    0.7043330                  0.7136764    0.2627957
Lag 1280                    0.6637319                  0.6861916    0.2217190
Chain 6 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 256  0.6926333          0.6039906         0.8901740
Lag 512  0.4971088          0.4152913         0.8220589
Lag 768  0.3761965          0.3067711         0.7766692
Lag 1024 0.2988411          0.2566547         0.7415017
Lag 1280 0.2382502          0.2171246         0.7074865
         b1nodematch.b1_gender.female b1nodematch.b1_gender.male b1cov.b1_age
Lag 0                       1.0000000                  1.0000000    1.0000000
Lag 256                     0.8912129                  0.8701079    0.6807525
Lag 512                     0.8136554                  0.7904833    0.4783346
Lag 768                     0.7591598                  0.7336232    0.3583303
Lag 1024                    0.7222278                  0.6832771    0.2852755
Lag 1280                    0.6862041                  0.6392340    0.2267765
Chain 7 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 256  0.6720099          0.5578472         0.8870763
Lag 512  0.4627328          0.3581277         0.8161864
Lag 768  0.3461662          0.2541563         0.7694078
Lag 1024 0.2569238          0.1915337         0.7319101
Lag 1280 0.2075656          0.1470170         0.7025771
         b1nodematch.b1_gender.female b1nodematch.b1_gender.male b1cov.b1_age
Lag 0                       1.0000000                  1.0000000    1.0000000
Lag 256                     0.8900874                  0.8746249    0.6650682
Lag 512                     0.8107650                  0.7951650    0.4524285
Lag 768                     0.7545832                  0.7384347    0.3362494
Lag 1024                    0.7115117                  0.6978314    0.2459340
Lag 1280                    0.6745173                  0.6647041    0.1948223
Chain 8 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 256  0.6551616          0.5778897         0.8724544
Lag 512  0.4484452          0.3797746         0.7992457
Lag 768  0.3154576          0.2779795         0.7491711
Lag 1024 0.2344501          0.2165930         0.7117201
Lag 1280 0.1779218          0.1735995         0.6855801
         b1nodematch.b1_gender.female b1nodematch.b1_gender.male b1cov.b1_age
Lag 0                       1.0000000                  1.0000000    1.0000000
Lag 256                     0.8991669                  0.8585259    0.6452315
Lag 512                     0.8303019                  0.7687964    0.4381757
Lag 768                     0.7778521                  0.7011824    0.3085934
Lag 1024                    0.7348994                  0.6497475    0.2268542
Lag 1280                    0.6956465                  0.6111093    0.1714640
Chain 9 
             edges gwb1deg.fixed.0.15 gwb1dsp.fixed.0.5
Lag 0    1.0000000          1.0000000         1.0000000
Lag 256  0.6609134          0.5836556         0.8877079
Lag 512  0.4689351          0.3889644         0.8218640
Lag 768  0.3502866          0.2794329         0.7793747
Lag 1024 0.2827982          0.2258947         0.7512313
Lag 1280 0.2479635          0.1938155         0.7291055
         b1nodematch.b1_gender.female b1nodematch.b1_gender.male b1cov.b1_age
Lag 0                       1.0000000                  1.0000000    1.0000000
Lag 256                     0.8905888                  0.8609453    0.6560757
Lag 512                     0.8187634                  0.7674188    0.4593463
Lag 768                     0.7667055                  0.7035730    0.3437351
Lag 1024                    0.7275068                  0.6560772    0.2737331
Lag 1280                    0.6925682                  0.6204053    0.2391251

Sample statistics burn-in diagnostic (Geweke):
Chain 1 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                       edges           gwb1deg.fixed.0.15 
                  0.83504894                   1.02994287 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                 -0.87076550                   0.02235246 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                 -0.89266760                   0.79173345 

Individual P-values (lower = worse):
                       edges           gwb1deg.fixed.0.15 
                   0.4036902                    0.3030368 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                   0.3838822                    0.9821668 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                   0.3720352                    0.4285161 
Joint P-value (lower = worse):  0.4889494 
Chain 2 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                       edges           gwb1deg.fixed.0.15 
                  1.39750009                   2.03248984 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                 -0.32832051                  -0.28110014 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                 -0.09268892                   1.50740734 

Individual P-values (lower = worse):
                       edges           gwb1deg.fixed.0.15 
                  0.16226324                   0.04210409 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                  0.74266934                   0.77863359 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                  0.92615070                   0.13170628 
Joint P-value (lower = worse):  0.2585445 
Chain 3 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                       edges           gwb1deg.fixed.0.15 
                   0.8181692                    1.0730253 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                   0.5310449                   -0.6778692 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                   1.5184877                    0.6157182 

Individual P-values (lower = worse):
                       edges           gwb1deg.fixed.0.15 
                   0.4132606                    0.2832598 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                   0.5953877                    0.4978546 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                   0.1288915                    0.5380805 
Joint P-value (lower = worse):  0.4156927 
Chain 4 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                       edges           gwb1deg.fixed.0.15 
                   1.0648936                    0.8410075 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                   0.6475327                    0.2322644 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                   0.5873750                    0.9409822 

Individual P-values (lower = worse):
                       edges           gwb1deg.fixed.0.15 
                   0.2869241                    0.4003437 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                   0.5172872                    0.8163326 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                   0.5569519                    0.3467140 
Joint P-value (lower = worse):  0.3645011 
Chain 5 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                       edges           gwb1deg.fixed.0.15 
                   -2.676119                    -2.638051 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                   -3.431800                    -1.792745 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                   -1.406467                    -2.329736 

Individual P-values (lower = worse):
                       edges           gwb1deg.fixed.0.15 
                0.0074480166                 0.0083384120 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                0.0005995898                 0.0730136756 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                0.1595855783                 0.0198201057 
Joint P-value (lower = worse):  0.2759939 
Chain 6 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                       edges           gwb1deg.fixed.0.15 
                  -1.8589085                   -2.8866123 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                  -3.3838284                   -0.9250280 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                  -0.6631141                   -2.0353891 

Individual P-values (lower = worse):
                       edges           gwb1deg.fixed.0.15 
                0.0630401036                 0.0038941369 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                0.0007148265                 0.3549513438 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                0.5072574908                 0.0418117383 
Joint P-value (lower = worse):  0.09459417 
Chain 7 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                       edges           gwb1deg.fixed.0.15 
                   0.6157156                    0.6361842 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                   0.3312007                   -0.1279171 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                   0.5937297                    0.4945037 

Individual P-values (lower = worse):
                       edges           gwb1deg.fixed.0.15 
                   0.5380823                    0.5246564 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                   0.7404929                    0.8982146 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                   0.5526929                    0.6209505 
Joint P-value (lower = worse):  0.5317512 
Chain 8 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                       edges           gwb1deg.fixed.0.15 
                  -0.4049230                   -0.8617011 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                  -0.4860840                   -1.8919772 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                   0.5750310                   -0.4015643 

Individual P-values (lower = worse):
                       edges           gwb1deg.fixed.0.15 
                  0.68553409                   0.38885200 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                  0.62690762                   0.05849402 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                  0.56527036                   0.68800468 
Joint P-value (lower = worse):  0.5210174 
Chain 9 

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5 

                       edges           gwb1deg.fixed.0.15 
                   0.4825563                    0.2475345 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                  -0.6165444                   -0.2711318 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                  -0.7127885                    0.5131922 

Individual P-values (lower = worse):
                       edges           gwb1deg.fixed.0.15 
                   0.6294108                    0.8044946 
           gwb1dsp.fixed.0.5 b1nodematch.b1_gender.female 
                   0.5375353                    0.7862897 
  b1nodematch.b1_gender.male                 b1cov.b1_age 
                   0.4759766                    0.6078169 
Joint P-value (lower = worse):  0.7595828 

Note: To save space, only one in every 16 iterations of the MCMC sample
  used for estimation was stored for diagnostics. Sample size per chain
  was originally around 64032 with thinning interval 16.

Note: MCMC diagnostics shown here are from the last round of
  simulation, prior to computation of final parameter estimates.
  Because the final estimates are refinements of those used for this
  simulation run, these diagnostics may understate model performance.
  To directly assess the performance of the final model on in-model
  statistics, please use the GOF command: gof(ergmFitObject,
  GOF=~model).

References

Abbasi, K., Alam, A., Brohi, N. A., Brohi, I. A., & Nasim, S. (2021). P2P lending fintechs and SMEs’ access to finance. Economics Letters, 204, 109890. https://doi.org/10.1016/j.econlet.2021.109890
Aliano, M., Alnabulsi, K., Cestari, G., & Ragni, S. (2023). The role of gender and education in peer-to-peer lending activities: Evidence from a european cross-country study. European Scientific Journal ESJ, 2. https://doi.org/10.19044/esipreprint.2.2023.p95
Cooper, D., Gorbachev, O., & Luengo‐Prado, M. J. (2023). Consumption, credit, and the missing young. Journal of Money, Credit and Banking, 55(2), 379–405.
Lim, K. B., Yeo, S. F., Lo, C., & Tan, C. L. (2023). Understanding of peer-to-peer lending platform intention: Evidence among millennials.
Liu, Y., Baals, L. J., Osterrieder, J., & Hadji-Misheva, B. (2024). Network centrality and credit risk: A comprehensive analysis of peer-to-peer lending dynamics. Finance Research Letters, 63, 105308. https://doi.org/10.1016/j.frl.2024.105308
Serrano-Cinca, C., Gutiérrez-Nieto, B., & López-Palacios, L. (2015). Determinants of default in P2P lending. PLOS ONE, 10(10), e0139427. https://doi.org/10.1371/journal.pone.0139427